K-Theory of $C^{*}(X)$ I'm new to K-Theory for $C^{*}$-algebra and $C^{*}$-algebra of groups.
If $X$ is the group of finite support bijections of natural numbers then what is the K-Theory of $C^{*}(X)$?
I was planning to calculate $C^{*}(X)$ first and then its projections and their homotopy classes but I failed to determine $C^{*}(X)$ in the first place.
Here is the definition of the $C^{*}$-algebra of a group:
https://pages.uoregon.edu/ncp/Courses/2016ShanghaiCrPrdFiniteGps/Slides/Lecture1_Print_NoP.pdf
Can anybody help me?
Thanks a lot.
 A: The group you describe should be the infinite symmetric group $S_{\infty}$. The $K$-theory of its $C^*$-algebra has been determined by Kerov and Vershik in
The K -functor (Grothendieck group) of the infinite symmetric group
https://link.springer.com/article/10.1007/BF02104985
The main result can be summarised as follows: Let $\mathcal{A}$ be the ring of symmetric polynomials in infinitely many variables. This is isomorphic to $\mathbb{Z}[a_1, a_2, \dots]$, where $a_i$ is the $i$th elementary symmetric function in an infinite number of arguments. Then
$$
K_0(C^*S_{\infty}) \cong \mathcal{A}\,/\, (a_1 - 1)\mathcal{A}\ .
$$
The isomorphism sends the irreducible representation $\pi_{\lambda}$ corresponding to the Young diagram $\lambda$ to the Schur function corresponding to $\lambda$. This is also an isomorphism of rings, where the ring structure on the left hand side comes from the observation that
$$
K_0(C^*S_{\infty}) \cong \lim_n K_0(C^*S_n)
$$
and multiplies two representations $\pi_1 \colon S_n \to GL(V)$ and $\pi_1 \colon S_m \to GL(W)$ to
$$
Ind_{S_n \times S_m}^{S_{n+m}}(\pi_1 \otimes \pi_2)\ ,
$$
where $S_n \times S_m$ sits inside $S_{n+m}$ with $S_n$ permuting the first $n$ elements and $S_m$ permuting the other $m$.
